A subsingleton generally refers to a subset (of some ambient set AA) having at most one element. That is, it is a subset BB of AA such that any two elements of BB are equal.

Of course, classically any subsingleton is either empty or a singleton, but constructively this need not hold. In a topos, the “object of subsingletons in AA” is the partial map classifier for AA, often denoted A A_\bot.


Sometimes a slightly different convention is used: There what we call subsingletons are called subterminals, and a subset BB of AA is a subsingleton if and only if there exists an element aAa \in A such that every element of BB is equal to aa. With this nomenclature, any subsingleton is a subterminal, but the converse doesn’t hold in general. (See flabby sheaf for a class of examples where the converse does hold.)

Revised on May 9, 2016 15:23:11 by Urs Schreiber (